Question: Simplify and expand the following expression: $ \dfrac{4}{5k + 10}+ \dfrac{2}{k - 2}- \dfrac{3k}{k^2 - 4} $
First find a common denominator by finding the least common multiple of the denominators. Try factoring the denominators. We can factor a $5$ out of denominator in the first term: $ \dfrac{4}{5k + 10} = \dfrac{4}{5(k + 2)}$ We can factor the quadratic in the third term: $ \dfrac{3k}{k^2 - 4} = \dfrac{3k}{(k + 2)(k - 2)}$ Now we have: $ \dfrac{4}{5(k + 2)}+ \dfrac{2}{k - 2}- \dfrac{3k}{(k + 2)(k - 2)} $ The least common multiple of the denominators is: $ 5(k + 2)(k - 2)$ In order to get the first term over $5(k + 2)(k - 2)$ , multiply by $\dfrac{k - 2}{k - 2}$ $ \dfrac{4}{5(k + 2)} \times \dfrac{k - 2}{k - 2} = \dfrac{4(k - 2)}{5(k + 2)(k - 2)} $ In order to get the second term over $5(k + 2)(k - 2)$ , multiply by $\dfrac{5(k + 2)}{5(k + 2)}$ $ \dfrac{2}{k - 2} \times \dfrac{5(k + 2)}{5(k + 2)} = \dfrac{10(k + 2)}{5(k + 2)(k - 2)} $ In order to get the third term over $5(k + 2)(k - 2)$ , multiply by $\dfrac{5}{5}$ $ \dfrac{3k}{(k + 2)(k - 2)} \times \dfrac{5}{5} = \dfrac{15k}{5(k + 2)(k - 2)} $ Now we have: $ \dfrac{4(k - 2)}{5(k + 2)(k - 2)} + \dfrac{10(k + 2)}{5(k + 2)(k - 2)} - \dfrac{15k}{5(k + 2)(k - 2)} $ $ = \dfrac{ 4(k - 2) + 10(k + 2) - 15k} {5(k + 2)(k - 2)} $ Expand: $ = \dfrac{4k - 8 + 10k + 20 - 15k}{5k^2 - 20} $ $ = \dfrac{-k + 12}{5k^2 - 20}$